Remarks on Kato's inequality when ∆_pu is a measure

抄録

Let Ω be a bounded domain of R^N (N ≥ 1). In this article, we shall study Kato'’s inequality when ∆_pu is a measure, where ∆_pu denotes a p-Laplace operator with 1 < p < ∞. The classical Kato'’s inequality for a Laplacian asserts that given any function u ∈ L¹_loc(Ω) such that ∆u ∈ L¹_loc(Ω), then ∆(u⁺) is a Radon measure and the following holds: ∆(u⁺) ≥ χ_{[u ≥ 0]}∆u in D^′(Ω). Our main result extends Kato’'s inequality to the case where ∆_pu is a Radon measures on Ω. We also establish the inverse maximum principle for ∆_p.